I have a WMS layer in QGIS and I want to select and extract certain features in order to vectorize them and perform clipping with other vector layers. Is there any way of doing this? I don't have access to the WMS source files.
No you can't (in any sensible way). WMS services serve images not data (see here) to the user. While you could use some fancy image processing software to edit the images I don't think that's what you're after. There is almost certainly a better way to achieve whatever objective you're aiming at.
One way or another you're going to want to get hold of the vector data which has been used to create the map you're working from. If it's open data that ought to be easy. If it's not, then you're going to need to think about copyright anyway - and there's probably a better solution by looking instead at a source of open data, or a source of closed data which you have the right to use.
There are some more advanced options with WMS which are beyond my knowledge - but they might offer some workarounds if you're stuck with using a particular WMS service. See this question.
To add a bit to the former answer,
some ArcGIS services serve WFS as well as WMS, this is visible in QGIS.
This is not supposed to happen, but it never hurts to test if the WMS you want to access has that capability.
you can try adding the service using the data source manager's arcgis wfs using the same address. i.e. using a url ending in mapserver and not featureserver.
I wrote a short guide about that here along with the note about ArcGIS services and some examples
(https://ags.iplan.gov.il/arcgis/rest/services/PlanningPublic/Xplan/MapServer for example works that way even though they set it up as only a map service)
as stated above you should always mind the license under which the data provided.